Hurewicz theorem

In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

Statement of the theorems

The Hurewicz theorems are a key link between homotopy groups and homology groups.

Absolute version

For any path-connected space X and positive integer n there exists a group homomorphism

${\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X),}$

called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator ${\displaystyle u_{n}\in H_{n}(S^{n})}$, then a homotopy class of maps ${\displaystyle f\in \pi _{n}(X)}$ is taken to ${\displaystyle f_{*}(u_{n})\in H_{n}(X)}$.

The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism.

• For ${\displaystyle n\geq 2}$, if X is ${\displaystyle (n-1)}$-connected (that is: ${\displaystyle \pi _{i}(X)=0}$ for all ${\displaystyle i<n}$), then ${\displaystyle {\tilde {H_{i}}}(X)=0}$ for all ${\displaystyle i<n}$, and the Hurewicz map ${\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X)}$ is an isomorphism.^{[1]: 366, Thm.4.32} This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map ${\displaystyle h_{*}\colon \pi _{n+1}(X)\to H_{n+1}(X)}$ is an epimorphism in this case.^{[1]: 390, ?}
• For ${\displaystyle n=1}$, the Hurewicz homomorphism induces an isomorphism ${\displaystyle {\tilde {h}}_{*}\colon \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\to H_{1}(X)}$, between the abelianization of the first homotopy group (the fundamental group) and the first homology group.

Relative version

For any pair of spaces ${\displaystyle (X,A)}$ and integer ${\displaystyle k>1}$ there exists a homomorphism

${\displaystyle h_{*}\colon \pi _{k}(X,A)\to H_{k}(X,A)}$

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both ${\displaystyle X}$ and ${\displaystyle A}$ are connected and the pair is ${\displaystyle (n-1)}$-connected then ${\displaystyle H_{k}(X,A)=0}$ for ${\displaystyle k<n}$ and ${\displaystyle H_{n}(X,A)}$ is obtained from ${\displaystyle \pi _{n}(X,A)}$ by factoring out the action of ${\displaystyle \pi _{1}(A)}$. This is proved in, for example, Whitehead (1978) by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by Brown & Higgins (1981) as a statement about the morphism

${\displaystyle \pi _{n}(X,A)\to \pi _{n}(X\cup CA),}$

where ${\displaystyle CA}$ denotes the cone of ${\displaystyle A}$. This statement is a special case of a homotopical excision theorem, involving induced modules for ${\displaystyle n>2}$ (crossed modules if ${\displaystyle n=2}$), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.

Triadic version

For any triad of spaces ${\displaystyle (X;A,B)}$ (i.e., a space X and subspaces A, B) and integer ${\displaystyle k>2}$ there exists a homomorphism

${\displaystyle h_{*}\colon \pi _{k}(X;A,B)\to H_{k}(X;A,B)}$

from triad homotopy groups to triad homology groups. Note that

${\displaystyle H_{k}(X;A,B)\cong H_{k}(X\cup (C(A\cup B))).}$

The Triadic Hurewicz Theorem states that if X, A, B, and ${\displaystyle C=A\cap B}$ are connected, the pairs ${\displaystyle (A,C)}$ and ${\displaystyle (B,C)}$ are ${\displaystyle (p-1)}$-connected and ${\displaystyle (q-1)}$-connected, respectively, and the triad ${\displaystyle (X;A,B)}$ is ${\displaystyle (p+q-2)}$-connected, then ${\displaystyle H_{k}(X;A,B)=0}$ for ${\displaystyle k<p+q-2}$ and ${\displaystyle H_{p+q-1}(X;A)}$ is obtained from ${\displaystyle \pi _{p+q-1}(X;A,B)}$ by factoring out the action of ${\displaystyle \pi _{1}(A\cap B)}$ and the generalised Whitehead products. The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental ${\displaystyle \operatorname {cat} ^{n}}$-group of an n-cube of spaces.

Simplicial set version

The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition.^[2]

Rational Hurewicz theorem

Rational Hurewicz theorem:^[3][4] Let X be a simply connected topological space with ${\displaystyle \pi _{i}(X)\otimes \mathbb {Q} =0}$ for ${\displaystyle i\leq r}$. Then the Hurewicz map

${\displaystyle h\otimes \mathbb {Q} \colon \pi _{i}(X)\otimes \mathbb {Q} \longrightarrow H_{i}(X;\mathbb {Q} )}$

induces an isomorphism for ${\displaystyle 1\leq i\leq 2r}$ and a surjection for ${\displaystyle i=2r+1}$.

Notes

1. Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79160-1
2. Goerss, Paul G.; Jardine, John Frederick (1999), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1, III.3.6, 3.7
3. Klaus, Stephan; Kreck, Matthias (2004), "A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres", Mathematical Proceedings of the Cambridge Philosophical Society, 136 (3): 617–623, Bibcode:2004MPCPS.136..617K, doi:10.1017/s0305004103007114, S2CID 119824771
4. Cartan, Henri; Serre, Jean-Pierre (1952), "Espaces fibrés et groupes d'homotopie, II, Applications", Comptes rendus de l'Académie des Sciences, 2 (34): 393–395

References

• Brown, Ronald (1989), "Triadic Van Kampen theorems and Hurewicz theorems", Algebraic topology (Evanston, IL, 1988), Contemporary Mathematics, vol. 96, Providence, RI: American Mathematical Society, pp. 39–57, doi:10.1090/conm/096/1022673, ISBN 9780821851029, MR 1022673
• Brown, Ronald; Higgins, P. J. (1981), "Colimit theorems for relative homotopy groups", Journal of Pure and Applied Algebra, 22: 11–41, doi:10.1016/0022-4049(81)90080-3, ISSN 0022-4049
• Brown, R.; Loday, J.-L. (1987), "Homotopical excision, and Hurewicz theorems, for n-cubes of spaces", Proceedings of the London Mathematical Society, Third Series, 54: 176–192, CiteSeerX 10.1.1.168.1325, doi:10.1112/plms/s3-54.1.176, ISSN 0024-6115
• Brown, R.; Loday, J.-L. (1987), "Van Kampen theorems for diagrams of spaces", Topology, 26 (3): 311–334, doi:10.1016/0040-9383(87)90004-8, ISSN 0040-9383

• Rotman, Joseph J. (1988), An Introduction to Algebraic Topology, Graduate Texts in Mathematics, vol. 119, Springer-Verlag (published 1998-07-22), ISBN 978-0-387-96678-6
• Whitehead, George W. (1978), Elements of Homotopy Theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, ISBN 978-0-387-90336-1